Chen–Zhu conjecture on primes in the gaps of a numerical semigroup

Let $G_0(a,b)$ be the set of gaps of the numerical semigroup generated by $a$ and $b$, let $g_{0,a,b}$ be its Frobenius number, and let $\pi^*_{0,a,b}$ be the number of primes in $G_0(a,b)$. Let $\pi(x)$ denote the number of primes not exceeding $x$. Chen–Zhu conjecture. For coprime integers $b>a\geq1$,

$$\pi^*_{0,a,b}\geq\frac{1}{2}\pi(g_{0,a,b}),$$

with equality if and only if one of the following holds:

$$(1)\ a=1;\qquad (2)\ (a,b)=(2,3);\qquad (3)\ (a,b)=(2,5);\qquad (4)\ (a,b)=(3,5).$$

The conjecture concerns the distribution of primes among the gaps of a two-generated numerical semigroup. The supplied text gives no resolution status for it, so it is recorded as open.